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A178685 G.f.: A(x) = x/Series_Reversion(G(x)) where x*A(G(x)) = G(x) is the g.f. of A000699. 2
1, 1, 3, 17, 143, 1569, 20911, 324673, 5720319, 112365569, 2430174335, 57327084033, 1464259637503, 40255921184257, 1185312426797823, 37219582236845057, 1241715255479353343, 43867539372405948417, 1636223622007966705663, 64261630682976366944257 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

The limit of (a(n+1)/a(n))/n = 2. - Paul D. Hanna, Aug 08 2014

The g.f. of related sequence A000699, where A000699(n) is the number of irreducible diagrams with 2n nodes, satisfies: G(x) = x + x^2*[d/dx G(x)^2/x].

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..300

FORMULA

G.f. satisfies:

(1) [x^(n+1)] A(x)^(n+2) = 2(n+1)*[x^n] A(x)^(n+2).

(2) A(x) = 1 + x*A(x) * (A(x) + x*A'(x)) / (A(x) - x*A'(x)). - Paul D. Hanna, Aug 08 2014

EXAMPLE

G.f.: A(x) = 1 + x + 3*x^2 + 17*x^3 + 143*x^4 + 1569*x^5 + 20911*x^6 +...

A(x) satisfies: A(G(x)) = G(x)/x where G(x) is the g.f. of A000699:

G(x) = x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2830*x^6 + 38232*x^7 +...

which in turn satisfies: G(x) = x + x^2*[d/dx G(x)^2/x].

...

The table of coefficients in the initial powers of g.f. A(x) begins:

A^1: [1, 1,  3,  17,  143,  1569,  20911,  324673,  5720319, ...];

A^2: [1, 2,  7,  40,  329,  3526,  46107,  705444, 12289245, ...];

A^3: [1, 3, 12,  70,  567,  5949,  76350, 1150920, 19818801, ...];

A^4: [1, 4, 18, 108,  867,  8928, 112524, 1671000, 28436085, ...];

A^5: [1, 5, 25, 155, 1240, 12566, 155650, 2277050, 38285275, ...];

A^6: [1, 6, 33, 212, 1698, 16980, 206902, 2982096, 49529871, ...];

A^7: [1, 7, 42, 280, 2254, 22302, 267624, 3801036, 62355195, ...];

A^8: [1, 8, 52, 360, 2922, 28680, 339348, 4750872, 76971171, ...];

...

In the above table, notice that the main diagonal:

[1, 2, 12, 108, 1240, 16980, 267624, 4750872, 93615408, ...]

is related to the secondary diagonal in the following way:

[2*1, 4*3, 6*18, 8*155, 10*1698, 12*22302, 14*339348, 16*5850963, ...].

PROG

(PARI) {a(n)=local(A=[1, 1]); for(i=2, n, A=concat(A, 0); A[#A]=(2*(#A-1)*Vec(Ser(A)^(#A))[#A-1]-Vec(Ser(A)^(#A))[#A])/(#A)); A[n+1]}

for(n=0, 25, print1(a(n), ", "))

(PARI) {a(n)=local(A=1+x); for(i=1, n, A = 1 + x*A*(A + x*A')/(A - x*A' +x*O(x^n))); polcoeff(A, n)}

for(n=0, 25, print1(a(n), ", "))

CROSSREFS

Cf. A000699.

Sequence in context: A001865 A189001 A087885 * A268254 A198860 A298691

Adjacent sequences:  A178682 A178683 A178684 * A178686 A178687 A178688

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Dec 31 2010

STATUS

approved

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Last modified January 17 14:12 EST 2019. Contains 319225 sequences. (Running on oeis4.)